Count a solid by complete layers of unit cubes

model. A rectangular prism is 4 cubes long, 3 cubes wide and 2 cubes high. How many unit cubes fill it without gaps?

1. Build or draw one complete base layer. Four cubes along the length and three along the width make twelve cube positions.
2. Place one cube in every position. The layer must be full; gaps would mean that a rectangular-prism formula no longer describes the actual packing.
3. Stack a second identical layer with no overlap or gap. The height of two cubes counts two complete layers of twelve.
4. Record 4 × 3 × 2 = 24 cubic units. Distinguish the base's area in square units from the solid's volume in cubic units.

Worked example: 24 unit cubes, so its volume is 24 cubic units. Each complete layer contains 4 × 3 = 12 cubes, and two layers contain 12 × 2.

p1. A prism is 3 unit cubes long, 2 wide and 4 high. Find its volume in cubic units.

Answer: 24

Teacher response: Six cubes per layer multiplied by four layers gives twenty-four.

p2. A filled prism has 15 cubes in each layer and 3 complete layers. Find its volume in cubic units.

Answer: 45

Teacher response: Three identical groups of fifteen cubes fill the prism.

p3. A prism contains 36 cubes in layers of 9 cubes each. How many layers are there?

Answer: 4

Teacher response: Four complete layers of nine account for all thirty-six cubes.

e1. A filled rectangular prism is 5 unit cubes long, 2 wide and 3 high. Find its volume and explain each factor.

Answer: 30

Look for:

• Describes a complete layer and the number of layers.
• Names cubic units and includes hidden cube positions.

Teacher response: Five times two gives ten cubes per layer; three layers give thirty.

transfer. If a 4-by-3-by-2 prism is turned so it stands 4 cubes high on a 3-by-2 base, does its volume change?

Answer: No cubes are added or removed. The layer description changes from 12 × 2 to 6 × 4, but both count 24 cubic units.

teach-1. 3 minutes: Build one layer and have learners label the two counts that describe its rows.

teach-2. 4 minutes: Stack or draw identical layers, then explain what the height counts.

teach-3. 5 minutes: Complete the practice with a labelled layer sketch alongside each multiplication.

teach-4. 3 minutes: Use the independent check and ask learners to explain how their method includes hidden cubes.

misconception. Count only the cubes visible on the outside of a drawing.

Teacher response: A solid can contain cubes hidden behind or underneath others. Counting full layers accounts for those positions as well as the visible ones.

support. Separate the solid into physical layers and count one layer before stacking them.

extend. Remove one unit cube from the packed prism. Explain why the actual volume is now 23 cubic units even though the outer bounding dimensions are unchanged.